Final answer:
The least common multiple (LCM) of 75 and 135 is 675, found by multiplying the highest powers of the prime factors, which are 3^3 and 5^2.
Step-by-step explanation:
To find the least common multiple (LCM) of 75 and 135, we can use the prime factorization method. First, let's find the prime factors of each number:
- 75 = 3 × 5 × 5
- 135 = 3 × 3 × 3 × 5
Next, identify the highest powers of the prime factors present in both numbers. For 75 and 135, they are:
- 3 (highest power in 135 is 3^3)
- 5 (highest power in 75 is 5^2)
Now, multiply these highest powers to find the LCM:
LCM = 3^3 × 5^2 = 27 × 25 = 675
Therefore, the LCM of 75 and 135 is 675.